[论文解读] Precisely computing bound orbits of spinning bodies around black holes I: General framework and results for nearly equatorial orbits

Lisa V. Drummond, Scott A. Hughes|arXiv (Cornell University)|Jan 31, 2022

Pulsars and Gravitational Waves Research参考文献 143被引用 41

一句话总结

本文提出一种频域方法，精确计算自旋物体在Kerr黑洞周围的束缚轨道，将自旋-曲率力纳入线性自旋阶次。结果表明，自旋会引发表征轨道频率Ωr和Ωφ的可观测偏移，这些偏移直接影响引力波的啁啾演化，对LISA观测中精确的EMRI波形建模至关重要。

ABSTRACT

Very large mass ratio binary black hole systems are of interest both as a clean limit of the two-body problem in general relativity, as well as for their importance as sources of low-frequency gravitational waves. At lowest order, the smaller body moves along a geodesic of the larger black hole's spacetime. Post-geodesic effects include the gravitational self force, which incorporates the backreaction of gravitational-wave emission, and the spin-curvature force, which arises from coupling of the small body's spin to the black hole's spacetime curvature. In this paper, we describe a method for precisely computing bound orbits of spinning bodies about black holes. Our analysis builds off of pioneering work by Witzany which demonstrated how to describe the motion of a spinning body to linear order in the small body's spin. Exploiting the fact that in the large mass-ratio limit spinning-body orbits are close to geodesics and using closed-form results due to van de Meent describing precession of the small body's spin along black hole orbits, we develop a frequency-domain formulation of the motion which can be solved very precisely. We examine a range of orbits with this formulation, focusing in this paper on orbits which are eccentric and nearly equatorial (i.e., the orbit's motion is $\mathcal{O}(S)$ out of the equatorial plane), but for which the small body's spin is arbitrarily oriented. We discuss generic orbits with general small-body spin orientation in a companion paper. We characterize the behavior of these orbits and show how the small body's spin shifts the frequencies $\Omega_r$ and $\Omega_\phi$ which affect orbital motion. These frequency shifts change accumulated phases which are direct gravitational-wave observables, illustrating the importance of precisely characterizing these quantities for gravitational-wave observations. (Abridged)

研究动机与目标

• 开发适用于极端质量比极限下自旋物体束缚轨道的高精度计算框架。
• 在小质量物体绕Kerr黑洞运动时，将自旋-曲率力纳入考虑，有效至小质量物体自旋的一阶展开。
• 表征自旋如何改变轨道频率Ωr和Ωφ，这些频率是引力波信号中的关键可观测量。
• 实现极端质量比旋进（EMRI）波形的精确建模，以支持LISA的探测能力。
• 为后续论文中对任意自旋取向的通用自旋物体轨道建模奠定基础。

提出的方法

• 采用频域公式求解Kerr时空中自旋物体的运动方程。
• 利用在大质量比极限下，自旋物体轨道接近测地线的特性。
• 应用van de Meent的闭式结果，对测地线上自旋进动进行建模以描述自旋演化。
• 以自旋的一阶展开求解Mathisson-Papapetrou方程，包含自旋-曲率力。
• 聚焦于近赤道、具有任意自旋取向的椭圆轨道。
• 计算由自旋引起的轨道频率偏移Ωr和Ωφ，这些偏移影响引力波的累积相位。

实验结果

研究问题

• RQ1 自旋-曲率力如何改变自旋物体在Kerr黑洞近赤道轨道上的径向频率Ωr和方位频率Ωφ？
• RQ2 在极端质量比极限下，自旋引起的轨道运动修正量的大小和结构如何？
• RQ3 与测地线运动相比，任意自旋取向如何影响束缚轨道的进动和频率偏移？
• RQ4 频域方法在多大程度上可实现自旋物体轨道的高精度计算？
• RQ5 这些自旋引起的频率偏移在LISA观测相关的引力波啁啾演化中产生何种影响？

主要发现

• 自旋-曲率力在即使自旋较小时，也会引起束缚轨道的径向频率Ωr和方位频率Ωφ的可观测偏移。
• 这些频率偏移在引力波信号中可直接观测到，并在数千圈轨道运动中累积显著相位变化。
• 频率偏移的大小取决于小质量物体自旋相对于轨道平面和黑洞自旋方向的取向。
• 通过利用质量比的微小性及已知的自旋进动解，该方法实现了轨道的高精度计算。
• 该框架可实现超越测地线的轨道动力学精确建模，对EMRI模板开发至关重要。
• 结果表明，即使在一阶自旋展开下，自旋效应对LISA频段EMRI观测也非可忽略。